Numerical analysis of a baryon and its dilatation modes in holographic QCD

We investigate a baryon and its dilatation modes in holographic QCD based on the Sakai-Sugimoto model, which is expressed as a $1+4$ dimensional $\mathrm{U}(N_f)$ gauge theory in the flavor space. For spatially rotational symmetric systems, we apply a generalized version of the Witten Ansatz, and reduce $1+4$ dimensional holographic QCD into a $1+2$ dimensional Abelian Higgs theory in a curved space. In the reduced theory, the holographic baryon is described as a two-dimensional topological object of an Abrikosov vortex. We numerically calculate the baryon solution of holographic QCD using a fine and large lattice with spacing of 0.04 fm and size of 10 fm. Using the relation between the baryon size and the zero-point location of the Higgs field in the description with the Witten Ansatz, we investigate a various-size baryon through this vortex description. As time-dependent size-oscillation modes (dilatation modes) of a baryon, we numerically obtain the lowest excitation energy of 577 MeV and deduce the dilatational excitation of a nucleon to be the Roper resonance ${\mathrm{N}}^{*}(1440)$.

Figure 1

Vortex description for the single ground-state baryon: the Higgs field $\varphi (r,w)=(\mathrm{Re}[\varphi ],\mathrm{Im}[\varphi ])$ (upper) and the Abelian gauge field $a(r,w)=(a_1,a_2)$ (lower) in the Landau gauge. The Kaluza-Klein unit of $M_{\mathrm{KK}}=1$ is used. The Higgs field has a zero point at $(r,w)\simeq (2.4,0)$, and its winding number around the zero point is equal to the baryon number, that is, $B=1$.

Figure 2

Temporal component of the U(1) gauge field ${\widehat{A}}_0(r,w)$ in the ground-state holographic baryon in the $M_{\mathrm{KK}}=1$ unit. This U(1) gauge field ${\widehat{A}}_0$ directly interacts with the topological density ${\rho }_B$ and leads to a repulsive force between topological densities, like the Coulomb interaction in QED.

Figure 3

Topological density ${\rho }_B(r,w)$ (upper) and total energy density $\mathcal{E}(r,w)$ (lower) for the ground-state solution of a single holographic baryon in the $M_{\mathrm{KK}}=1$ unit. Both densities have a peak around $(r,w)=(0,0)$.

Figure 4

$r^2$-multiplied topological and energy densities: $4\pi r^2{\rho }_B(r,w)$ (upper) and $4\pi r^2\mathcal{E}(r,w)$ (lower) for the ground-state solution of a single holographic baryon in the $M_{\mathrm{KK}}=1$ unit. Both figures have a peak around $(r,w)\simeq (2,0)$ near the vortex center, which roughly controls the baryon size.

Figure 5

Baryon density ${\rho }_B(r)$ and total energy density $\mathcal{E}(r)$ for the ground-state baryon. The lower panel shows $4\pi r^2{\rho }_B(r)$ and $4\pi r^2\mathcal{E}(r)$, including the integral measure factor $r^2$. Despite a significant difference between ${\rho }_B(r)$ and $\mathcal{E}(r)$ around the origin, this difference is reduced by the $r^2$ multiplication. Here, the $M_{\mathrm{KK}}=1$ unit is taken.

Figure 6

$\mathrm{SU}(2{)}_f$ energy density ${\mathcal{E}}^{{\mathrm{SU}(2)}_f}(r)$ and U(1) energy density ${\mathcal{E}}^{\mathrm{U}(1)}(r)$ of the ground-state baryon, together with the total energy density $\mathcal{E}(r)$ and baryon density ${\rho }_B(r)$, denoted by the solid lines. The lower panel shows $r^2$-multiplied values, $4\pi r^2\mathcal{E}(r)$, $4\pi r^2{\mathcal{E}}^{{\mathrm{SU}(2)}_f}(r)$, $4\pi r^2{\mathcal{E}}^{\mathrm{U}(1)}(r)$ and $4\pi r^2{\rho }_B(r)$. Here, the $M_{\mathrm{KK}}=1$ unit is taken.

Figure 7

Static energy $V(R)$ of the baryon with various size $R$ in the $M_{\mathrm{KK}}=1$ unit. The symbol denotes numerical data of the baryon energy with different sizes. The line denotes a fit curve of the potential by a quadratic function. This potential has a minimum corresponding to the ground-state, and its value is consistent with the previous ground-state calculation.

Figure 8

${\rho }_B$-weighted radius $d_r(R)$ and $d_w(R)$ of the holographic baryon in the direction $r$ and $w$, respectively, as the function of the size parameter $R$ in the $M_{\mathrm{KK}}=1$ unit. The solid line denotes $d=R$, which is to be realized in the case of the ordinary four-dimensional YM theory ($h(w)=k(w)=1$) without the CS term. If the holographic baryon were described with the ’t Hooft solution, $d_r(R)=d_w(R)=R$ would be satisfied.

Figure 9

Duality breaking parameter ${\mathrm{\Delta }}_{\mathrm{DB}}$ for various baryon size $R$ in the $M_{\mathrm{KK}}=1$ unit. The functional form of ${\mathrm{\Delta }}_{\mathrm{DB}}(R)$ seems to be quadratic and it takes minimum at $R\simeq 2.2$.

Figure 10

Schematic figure for mass splitting of the holographic baryon in terms of the $1/N_c$ expansion. (The symbol * denotes the excitation mode.) The holographic baryon has $O(N_c^0)$ and $O(N_c^{-1})$ splitting, corresponding to dilatation and rotational effect, respectively. Therefore, the order of low-lying baryon mass is to be $\mathrm{N}<\mathrm{\Delta }<{\mathrm{N}}^{*}<{\mathrm{\Delta }}^{*}$ in term of the $1/N_c$ expansion.

Figure 11

The Higgs field $\varphi (r,w)$ (upper) and the Abelian gauge field $a(r,w)=(a_1,a_2)$ (lower) for the BPS instanton in the Landau gauge in the $M_{\mathrm{KK}}=1$ unit. For iterative improvement in the main sections, this configuration is used as the starting point, i.e., the initial configuration of the iteration.

Figure 12

The U(1) gauge field ${\widehat{A}}_0$ in the case of BPS instanton in the $M_{\mathrm{KK}}=1$ unit. This is obtained by solving Eq. (25) for the self-dual ’t Hooft instanton.

Figure 13

$r^2$-multiplied topological and energy densities, $4\pi r^2{\rho }_B(r)$ and $4\pi r^2\mathcal{E}(r)$, in the case of the BPS instanton configuration in the $M_{\mathrm{KK}}=1$ unit.

Figure 14

Static baryon energy $V^{\mathrm{BPS}}(R)$ calculated from the BPS configurations in Eq. (56). The solid line is a fitting curve of quadratic function.